Electricity and Magnetism

Electricity and Magnetism

Learning Objectives

Define electric charge and apply Coulomb's Law.
Understand the concept of electric fields and apply Gauss's Law.
Calculate electric potential and capacitance.
Analyze simple DC circuits using Ohm's Law and Kirchhoff's Rules.
Describe magnetic fields, the magnetic force on moving charges, and sources of magnetic fields.
Explain electromagnetic induction using Faraday's Law and Lenz's Law.

Electricity and Magnetism (Electromagnetism) govern the interactions between electrically charged particles. This field is foundational for electrical engineering, power transmission, and electronics.

Electric Charge

Electric Charge Concepts

Matter is composed of atoms, which contain positively charged protons and negatively charged electrons. The fundamental unit of charge ( $e$ ) is $1.602 \times 10^{-19}$ Coulombs (C).

Protons: $+e$
Electrons: $-e$

Charge ( $q$ ) is a conserved property. The total charge of an isolated system remains constant. Like charges repel; opposite charges attract.

Coulomb's Law

Coulomb's Law Concepts

The fundamental force between two point charges ( $q_1$ and $q_2$ ) separated by a distance ( $r$ ) is given by Coulomb's Law. It is mathematically identical in form to Newton's Law of Universal Gravitation, but the force can be attractive or repulsive.

Coulomb's Law

F = k \frac{|q_1 q_2|}{r^2} = \frac{1}{4\pi\epsilon_0} \frac{|q_1 q_2|}{r^2}

Where:

$F$ is the magnitude of the electrostatic force (in Newtons, N).
$k$ is Coulomb's constant ( $8.987 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ ).
$\epsilon_0$ is the permittivity of free space ( $8.854 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ ).

Interactive Simulation

Use this electric force model to see how charge signs, magnitudes, and separation control attraction or repulsion.

Interactive Physics Simulation

Coulomb's Law Electrostatic Simulator

Adjust the charges on two point particles and see how the electrostatic force scales with charge magnitudes and separation distance. Opposite charges attract; like charges repel.

Attracts

Charge q15 μC

Charge q2-5 μC

Distance r10 cm

Governing Formula

F = k_e \cdot \frac{|q_1 \cdot q_2|}{r^2}

Where electrostatic constant $k_e \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$. Force is proportional to charge product and inversely proportional to $r^2$.

Force magnitude (F)

22.47 N

Interaction Type

Attractive

\vec{E}

The Electric Field ( $\vec{E}$ ) Concepts

Instead of thinking of charges exerting forces directly on one another across empty space, we introduce the concept of an electric field. A charge creates an electric field in the space around it, and another charge placed in that field feels a force.

Electric Field ( $\vec{E}$ )

The electrostatic force per unit charge exerted on a positive test charge ( $q_0$ ) at a specific point in space. Electric field lines point away from positive charges and towards negative charges.

Electric Field

Calculates the electric field at a point.

\vec{E} = \frac{\vec{F}}{q_0}

Variables

Symbol	Description	Unit
$\vec{E}$	Electric field	N/C or V/m
$\vec{F}$	Electrostatic force	N
$q_0$	Positive test charge	C

Electric Field of a Point Charge

Calculates the electric field created by a single point charge.

\vec{E} = k \frac{q}{r^2} \hat{r}

Variables

Symbol	Description	Unit
$\vec{E}$	Electric field	N/C or V/m
$k$	Coulomb's constant	$N\cdot m^2/C^2$
$q$	Point charge	C
$r$	Distance from the charge	m
$\hat{r}$	Unit vector pointing radially outward from the charge	dimensionless

Gauss's Law

Gauss's Law Concepts

Gauss's Law is a powerful alternative to Coulomb's law for calculating electric fields of symmetric charge distributions (like spheres, cylinders, or infinite planes). It relates the electric flux ( $\Phi_E$ ) passing through a closed "Gaussian" surface to the total charge enclosed within that surface ( $q_{enc}$ ).

Gauss's Law

Relates the electric flux passing through a closed surface to the charge enclosed within it.

\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}

Variables

Symbol	Description	Unit
$\Phi_E$	Electric flux	$V\cdot m$
$\vec{E}$	Electric field	N/C or V/m
$d\vec{A}$	Differential area vector	$m^2$
$q_{enc}$	Total enclosed charge	C
$\epsilon_0$	Permittivity of free space	$C^2/(N\cdot m^2)$

V

Electric Potential ( $V$ ) Concepts

Just as a mass in a gravitational field has gravitational potential energy, a charge in an electric field has electric potential energy ( $U$ ). Because the electrostatic force is conservative, we can define a potential energy function.

It is often more useful to talk about energy per unit charge, which is the electric potential (commonly called Voltage).

Electric Potential ( $V$ )

The electric potential energy per unit charge at a point in an electric field. The SI unit is the Volt (V), where $1 \text{ V} = 1 \text{ J/C}$ .

Electric Potential

Calculates the electric potential energy per unit charge.

V = \frac{U}{q}

Variables

Symbol	Description	Unit
$V$	Electric potential	V
$U$	Electric potential energy	J
$q$	Charge	C

Electric Potential of a Point Charge

Calculates the electric potential created by a single point charge.

V = k \frac{q}{r}

Variables

Symbol	Description	Unit
$V$	Electric potential	V
$k$	Coulomb's constant	$N\cdot m^2/C^2$
$q$	Point charge	C
$r$	Distance from the charge	m

Electric Potential ( $V$ ) Concepts

The difference in potential between two points ( $\Delta V$ ) is the voltage. It represents the work done per unit charge by an external agent moving a positive test charge between those points.

Voltage (Potential Difference)

Calculates the difference in electric potential between two points.

\Delta V = V_f - V_i = \frac{\Delta U}{q} = - \int \vec{E} \cdot d\vec{s}

Variables

Symbol	Description	Unit
$\Delta V$	Voltage or potential difference	V
$V_f, V_i$	Final and initial electric potentials	V
$\Delta U$	Change in electric potential energy	J
$q$	Charge	C
$\vec{E}$	Electric field	V/m
$d\vec{s}$	Differential path element	m

C

Capacitance ( $C$ ) Concepts

A capacitor is a device (usually two conducting plates separated by an insulator) that stores electric charge and potential energy.

Capacitance ( $C$ )

The ratio of the magnitude of charge on either plate ( $Q$ ) to the potential difference ( $\Delta V$ ) between them. The SI unit is the Farad (F), where $1 \text{ F} = 1 \text{ C/V}$ .

Capacitance

Calculates the capacitance of a device.

C = \frac{Q}{\Delta V}

Variables

Symbol	Description	Unit
$C$	Capacitance	F
$Q$	Magnitude of charge on either plate	C
$\Delta V$	Potential difference between plates	V

Parallel-Plate Capacitor

Calculates the capacitance for a parallel-plate configuration.

C = \epsilon_0 \frac{A}{d}

Variables

Symbol	Description	Unit
$C$	Capacitance	F
$\epsilon_0$	Permittivity of free space	$F/m or C^2/(N\cdot m^2)$
$A$	Area of each plate	$m^2$
$d$	Separation distance between plates	m

Capacitance ( $C$ ) Concepts

The energy stored in a capacitor is $U = \frac{1}{2} C (\Delta V)^2$ .

Current and Resistance

Current and Resistance Concepts

When charges move continuously, they form an electric current.

Current ( $I$ )

The rate of flow of electric charge through a cross-sectional area. The SI unit is the Ampere (A), where $1 \text{ A} = 1 \text{ C/s}$ .

Electric Current

Calculates the rate of charge flow.

I = \frac{dq}{dt}

Variables

Symbol	Description	Unit
$I$	Current	A
$dq$	Differential amount of charge	C
$dt$	Differential time interval	s

Current and Resistance Concepts

By convention, the direction of current is the direction positive charges would flow (opposite to the actual flow of electrons).

To drive a current through a conductor, a potential difference ( $\Delta V$ ) is required. Most materials resist this flow.

Ohm's Law

Relates current, voltage, and resistance for ohmic materials.

I = \frac{\Delta V}{R}

Variables

Symbol	Description	Unit
$I$	Current	A
$\Delta V$	Potential difference or voltage	V
$R$	Resistance	$\Omega$

Interactive Simulation

Use this circuit model to connect voltage, resistance, current, and power in a simple DC series path.

Interactive Physics Simulation

Ohm's Law Circuit Simulator

Switch between series and parallel resistance. Current, power, electron-flow speed, and bulb brightness update immediately.

Supply voltage12 V

Resistance R16 Ω

Resistance R24 Ω

Governing Formulas

Ohm's Law

I = \frac{V}{R}

Equivalent Resistance (series)

R_{eq} = R_1 + R_2

Equivalent R

10.00 Ω

Current

1.20 A

Power

14.40 W

Current and Resistance Concepts

Resistance depends on the material's resistivity ( $\rho$ ) and geometry: $R = \rho L / A$ .

The rate at which energy is dissipated in a resistor (power) is $P = I \Delta V = I^2 R = (\Delta V)^2 / R$ .

Kirchhoff's Rules

Kirchhoff's Rules Concepts

For analyzing complex circuits, we use two rules based on conservation laws:

Junction Rule (Conservation of Charge): The sum of currents entering any junction must equal the sum of currents leaving that junction ( $\Sigma I_{in} = \Sigma I_{out}$ ).
Loop Rule (Conservation of Energy): The algebraic sum of changes in potential around any closed circuit path (loop) must be zero ( $\Sigma \Delta V = 0$ ).

Magnetism

Magnetism Concepts

Moving charges (currents) create magnetic fields ( $\vec{B}$ ). Magnetic fields, in turn, exert forces on moving charges.

Magnetic Force ( $\vec{F}_B$ )

The force exerted by a magnetic field $\vec{B}$ on a charge $q$ moving with velocity $\vec{v}$ . The direction is determined by the Right-Hand Rule.

Magnetic Force

Calculates the force exerted by a magnetic field on a moving charge.

\vec{F}_B = q(\vec{v} \times \vec{B})

Variables

Symbol	Description	Unit
$\vec{F}_B$	Magnetic force	N
$q$	Charge	C
$\vec{v}$	Velocity vector	m/s
$\vec{B}$	Magnetic field vector	T

Work and Magnetic Force

Notice that the force is always perpendicular to both the velocity and the magnetic field. Therefore, a magnetic field can change the direction of a moving charge but cannot do work on it (cannot change its kinetic energy).

Sources of Magnetic Fields

Sources of Magnetic Fields Concepts

Magnetic fields are created by currents. The Biot-Savart Law and Ampere's Law allow us to calculate these fields. For highly symmetric current distributions, Ampere's Law is especially useful.

Magnetic Field of a Long Straight Wire

Calculates the magnetic field magnitude at a distance r from a long, straight current-carrying wire.

B = \frac{\mu_0 I}{2\pi r}

Variables

Symbol	Description	Unit
$B$	Magnetic field magnitude	T
$\mu_0$	Permeability of free space	$4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$
$I$	Current in the wire	A
$r$	Perpendicular distance from the wire	m

Magnetic Field at the Center of a Circular Loop

Calculates the magnetic field magnitude at the center of a single circular current loop.

B = \frac{\mu_0 I}{2R}

Variables

Symbol	Description	Unit
$B$	Magnetic field magnitude	T
$\mu_0$	Permeability of free space	$4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$
$I$	Current in the wire loop	A
$R$	Radius of the circular loop	m

Magnetic Field Inside a Long Solenoid

Calculates the nearly uniform magnetic field inside a long coil of wire.

B = \mu_0 n I

Variables

Symbol	Description	Unit
$B$	Magnetic field magnitude	T
$\mu_0$	Permeability of free space	$4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$
$n$	Number of turns per unit length (N/L)	1/m
$I$	Current in the wire	A

Applications of Magnetism

Electric Motors: Using magnetic forces on current-carrying loops to generate torque.
Transformers: Utilizing electromagnetic induction to step-up or step-down AC voltages.
Magnetic Levitation (Maglev): Using magnetic repulsion to lift and propel trains, reducing friction.

Electromagnetic Induction

Electromagnetic Induction Concepts

A changing magnetic field can induce an electric current in a conductor. This is the principle behind all electrical generators.

Faraday's Law of Induction

Calculates the induced electromotive force in a closed loop.

\mathcal{E} = - \frac{d\Phi_B}{dt} = - \frac{d(\vec{B} \cdot \vec{A})}{dt} = - \frac{d(B A \cos\theta)}{dt}

Variables

Symbol	Description	Unit
$\mathcal{E}$	Induced electromotive force (EMF)	V
$\Phi_B$	Magnetic flux	Wb
$t$	Time	s
$\vec{B}$	Magnetic field	T
$\vec{A}$	Area vector	$m^2$
$\theta$	Angle between the magnetic field and the area normal	$rad or ^\circ$

Electromagnetic Induction Concepts

The negative sign represents Lenz's Law, which states that the induced current will flow in a direction such that its own magnetic field opposes the change in flux that caused it. This is a consequence of the conservation of energy.

Direct Current (DC) Circuits

Resistors in Series and Parallel

In DC circuits, components can be connected in two primary configurations, which dictate how voltage and current behave across them.

Series Circuits: Components are connected end-to-end, forming a single path. The current ( $I$ ) is identical through all components.
Parallel Circuits: Components are connected across the same two nodes, providing multiple paths. The voltage ( $\Delta V$ ) is identical across all parallel branches.

Equivalent Resistance in Series

Calculates the total resistance of multiple resistors connected in series.

R_{eq} = R_1 + R_2 + R_3 + \dots

Variables

Symbol	Description	Unit
$R_{eq}$	Equivalent series resistance	$\Omega$
$R_i$	Individual resistances	$\Omega$

Equivalent Resistance in Parallel

Calculates the total resistance of multiple resistors connected in parallel.

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots

Variables

Symbol	Description	Unit
$R_{eq}$	Equivalent parallel resistance	$\Omega$
$R_i$	Individual resistances	$\Omega$

Capacitance and Dielectrics

Dielectric Materials

When an insulating material (a dielectric) is placed between the plates of a capacitor, it increases the capacitance of the device by a dimensionless factor called the dielectric constant ( $\kappa$ ).

The molecules in the dielectric become polarized by the external electric field, creating an opposing internal electric field. This reduces the overall electric field between the plates, allowing the capacitor to store more charge for the same applied voltage.

Key Takeaways

Electric Charge is quantized and conserved. Like charges repel, opposites attract according to Coulomb's Law ( $F = kq_1q_2/r^2$ ).
An Electric Field ( $\vec{E} = \vec{F}/q$ ) is created by charges and exerts forces on other charges. Gauss's Law relates flux to enclosed charge.
Electric Potential ( $V = U/q$ ) is energy per unit charge (Voltage). Capacitors ( $C = Q/\Delta V$ ) store charge and energy.
Current ( $I = dq/dt$ ) is the flow of charge, driven by voltage and opposed by Resistance ( $V=IR$ , Ohm's Law).
Moving charges create Magnetic Fields ( $\vec{B}$ ), and magnetic fields exert forces on moving charges ( $\vec{F} = q\vec{v}\times\vec{B}$ ).
A changing magnetic flux induces an EMF (voltage) according to Faraday's Law ( $\mathcal{E} = -d\Phi_B/dt$ ), the basis of electrical generation.

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Learning Objectives

Electric Charge

Electric Charge Concepts

Coulomb's Law

Coulomb's Law Concepts

Coulomb's Law

Interactive Simulation

Coulomb's Law Electrostatic Simulator

The Electric Field (E⃗\vec{E}E)

The Electric Field (E⃗\vec{E}E) Concepts

Electric Field (E⃗\vec{E}E)

Electric Field

Electric Field of a Point Charge

Gauss's Law

Gauss's Law Concepts

Gauss's Law

Electric Potential (VVV)

Electric Potential (VVV) Concepts

Electric Potential (VVV)

Electric Potential

Electric Potential of a Point Charge

Electric Potential (VVV) Concepts

Voltage (Potential Difference)

Capacitance (CCC)

Capacitance (CCC) Concepts

Capacitance (CCC)

Capacitance

Parallel-Plate Capacitor

Capacitance (CCC) Concepts

Current and Resistance

Current and Resistance Concepts

Current (III)

Electric Current

Current and Resistance Concepts

Ohm's Law

Interactive Simulation

Ohm's Law Circuit Simulator

Current and Resistance Concepts

Kirchhoff's Rules

Kirchhoff's Rules Concepts

Magnetism

Magnetism Concepts

Magnetic Force (F⃗B\vec{F}_BFB​)

Magnetic Force

Work and Magnetic Force

Sources of Magnetic Fields

Sources of Magnetic Fields Concepts

Magnetic Field of a Long Straight Wire

Magnetic Field at the Center of a Circular Loop

Magnetic Field Inside a Long Solenoid

Applications of Magnetism

Electromagnetic Induction

Electromagnetic Induction Concepts

Faraday's Law of Induction

Electromagnetic Induction Concepts

Direct Current (DC) Circuits

Resistors in Series and Parallel

Equivalent Resistance in Series

Equivalent Resistance in Parallel

Capacitance and Dielectrics

Dielectric Materials

The Electric Field ( $\vec{E}$ )

The Electric Field ( $\vec{E}$ ) Concepts

Electric Field ( $\vec{E}$ )

Electric Potential ( $V$ )

Electric Potential ( $V$ ) Concepts

Electric Potential ( $V$ )

Electric Potential ( $V$ ) Concepts

Capacitance ( $C$ )

Capacitance ( $C$ ) Concepts

Capacitance ( $C$ )

Capacitance ( $C$ ) Concepts

Current ( $I$ )

Magnetic Force ( $\vec{F}_B$ )